J. Willard Gibbs (ca. 1870) derived a simple rule that determines the number of degrees of freedom for a heterogeneous system. If a system in thermodynamic equilibrium contains phases and components, then the phase rule states that the number of degrees of freedom is given by . Degrees of freedom represents the number of intensive variables (such as pressure, temperature, and composition) that can be varied arbitrarily over some finite range without changing the number of phases. The phase rule has been described as "pretty but powerful, qualitative yet exact".

This Demonstration considers only the simplest cases of one- and two-component systems. Omitted are such phenomena as multiple crystal structures, solid solutions, partial miscibility of liquids, and azeotrope formation. You can drag the locator over various regions of the phase diagrams. For one-component systems, this selects values for the pressure and temperature. For two-component systems, this selects the temperature and composition. The two-component phase diagram should actually be three-dimensional, with pressure providing an additional degree of freedom.

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Following is a compact derivation of the phase rule. At equilibrium, the chemical potential of each component , , …, is equal across each phase boundary, as is the temperature and pressure (taking account of hydrostatic effects, if necessary). This gives a total of variables. However these must conform to an equation of state in each phase. This leaves arbitrary intensive variables, which represents the number of degrees of freedom.

Remarkably, Gibbs' phase rule is isomorphic with Euler&CloseCurlyQuote;s formula , relating the number of vertices, edges, and faces of a simply-connected polyhedron.